#
Tag Archives: Mandelbrot

**123 × 321 = 39,483**

My interest in number multiplication goes back to exploring algorithms for generating Mandelbrot plots, which can require billions of multiplication operations on arbitrary precision numbers (numbers with *lots and lots* of digits).

**Multiplying** two numbers — calculating their *product* — is computationally intense because of the intermediate Cartesian product. Multiplying two 12-digit numbers creates a 24-digit result (12+12), but it *also* has an intermediate stage involving **144** (12×12) single digit multiplications.

Recently I learned an intriguing Japanese *visual* multiplication method.

Continue reading

23 Comments | tags: Cartesian product, Japanese multiplication method, Mandelbrot, multiplication | posted in Math

In recent posts I’ve presented the complex numbers and the complex plane. Those were just stepping stones to this post, which involves a basic fact about the Mandelbrot set. It’s something that I stumbled over recently (after tip-toeing around it many times, because math).

This is one of those places where something that seems complicated turns out to have a fairly simple (and kinda cool) way to see it when approached the right way. In this case, it’s the way multiplication *rotates* points on the complex plane. This allow us to actually visualize certain equations.

With that, we’re ready to move on to the “heart” of the matter…

Continue reading

18 Comments | tags: cartioid, complex numbers, fun with numbers, Mandelbrot, Mandelbrot fractal, real numbers | posted in Math

*Math version 1.0*

This image here of the **Mandelbrot fractal** might look like one of the uglier renderings you’ve seen, but it’s a thing of beauty to me. That’s because some code I wrote created it. Which, in itself, isn’t a deal (let alone a big one), but how that code works kind of is (at least for me).

The short version: the code implements special *virtual* math for calculating the Mandelbrot. That the image looks anything at all like it should shows the code works.

Yet according to that image, something wasn’t quite right.

Continue reading

7 Comments | tags: algorithm, Mandelbrot, Mandelbrot fractal, virtual reality | posted in Computers

So. **2020**. The start of a new decade. That’s just a bit surreal for me. I can remember wondering if **1984** would turn out anything like the novel. The future did turn out a bit like Orwell’s vision — it just took until **2016** or so to get there. It isn’t so much that Big Brother is watching (although, that too), but how our government corrupts and perverts facts and truth.

Good fiction is insightful about the human condition; good science fiction is insightful about our future. Over time, as advertised, the prescient film *Idiocracy* goes from SF comedy to anthropological documentary. Many others went from fiction to fact. (Fortunately, at least so far, *The Terminator* has not.)

Suffice to say, this year will seem surreal in more ways than one.

Continue reading

3 Comments | tags: 1984, 2001, 2001: A Space Odyssey, 2020, blog, blogging, George Orwell, Janus, Mandelbrot, Nineteen Eighty-Four, WordPress | posted in Life

*Complete Chaos!*

One of my favorite discoveries in life is the **Mandelbrot set**. Considering it gives me a strong sense of the numinous. I’ve been enthralled by it ever since Fractint, an MS-DOS program that generated fractals. I’ve posted about it a lot here; today I want to take you into the heart of its chaotic behavior.

The Mandelbrot set has a number of properties that make it such a fascinating study: Firstly, it demonstrates **chaos theory**. Secondly, it demonstrates how complex patterns can arise from simple beginnings. Thirdly, it reveals a problem concerning real numbers. Fourthly, every pixel is a demonstration of Turing’s Halting Problem. It’s also infinitely complex and incredibly beautiful.

Today we’re going to explore the shore of the Mandelbrot lake.

Continue reading

6 Comments | tags: chaos, chaos theory, fractals, Mandelbrot, Mandelbrot fractal | posted in Science

The **Thanksgiving holiday** we celebrate here in the USA has some unfortunate overtones regarding its colonial origin. Still, the idea of a festival of thanks is an ancient one — thanks for a good harvest or a good hunt. Or, in our case, thanks for helping us not die last winter.

As with Christmas or the Copenhagen interpretation, we tend to take a *“shut up and calculate!”* approach to the holidays. *“Shut up and shop!”* in the case of the Winter Solstice, and *“Shut up and give thanks!”* today.

One thing we can be very thankful about is patterns…

Continue reading

2 Comments | tags: atoms, DNA, electrons, entropy, Mandelbrot, Mandelbrot fractal, photons, quarks, Thanksgiving | posted in Life

I had thoughts about a second May Mandelbrot post that got a bit deeper into the weeds, but a couple attempts today went nowhere (except the trashcan). But I been having some fun exploring the **Mandelbrot** with Ultra Fractal, and I thought some pictures might be worth a few words.

Click on any to see bigger versions.

Continue reading

2 Comments | tags: Mandelbrot, Mandelbrot fractal | posted in Computers

Mandelbrot Antennae

*[click for big]*

I realized that, if I’m going to do the **Mandelbrot** in May, I’d better get a move on it. This ties to the main theme of Mind in May only in being about *computation* — but not about computation**alism** or consciousness. (Other than in the *subjective appreciation* of its sheer beauty.)
I’ve heard it called “the most complex” mathematical object, but that’s a hard title to earn, let alone hold. Its complexity does have attractive and fascinating aspects, though. For most, its visceral visual beauty puts it miles ahead of the cool intellectual poetry of Euler’s Identity (both beauties live on the same block, though).

For me, the cool thing about the Mandelbrot is that it’s a computation that can never be fully computed.

Continue reading

10 Comments | tags: computation, computer model, computer program, fractals, Mandelbrot, Mandelbrot fractal, mathematics, Turing Halting Problem | posted in Computers, Math

To start the last week of March Mathness, because it’s a Monday, I’m going to go easy on y’all with some light, easy topics. (Maybe I can lull you into paying attention for the major topic of the month: matrix rotation.)

It has occurred to me that, if I’m talking about math in March, I absolutely must mention one of my all-time favorite mathematical objects, the **Mandelbrot**. I’ll try to get to that today, but the main topic is a simple something that I ran into while working on my 3D model of the big island of Hawaii.

The question was: *How many miles are there per degree of ***latitude**?

Continue reading

10 Comments | tags: circle of latitude, cosine, Earth, latitude, longitude, Mandelbrot, Maths Town, nautical mile, radius | posted in Math

Back at the start of March Mathness I promised the math would be “fun” (really!), but anyone would be forgiven for thinking the previous two posts about Special Relativity weren’t all that much “fun.” (I really enjoy stuff like that, so it’s fun for *me*, but there’s no question it’s not everyone’s cup of tea.)

Trying to reach for something a bit lighter and potentially more appealing as the promised “fun,” I present, for your dining and dancing pleasure, a trio of number games that anyone can play and which might just tug at the corners of your enjoyment.

We can start with **277777788888899** (and why it’s special).

Continue reading

3 Comments | tags: 277777788888899, algorithm, calculation, Collatz conjecture, computation, Mandelbrot, mathematics, multiplicative persistence, Numberphile | posted in Math